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Exploring Exponential Functions: Graphing and Identifying Growth or Decay, Lecture notes of Mathematics

Mathematics

An introduction to exponential functions, explaining their definition, scientific applications, and the process of graphing them. The concept of horizontal asymptotes and the difference between exponential growth and decay based on the base value. Students will gain a better understanding of exponential functions and how to identify their graph's behavior.

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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bg1
GraphingExponentialFunctions
WhatisanExponentialFunction?
Exponentialfunctionsareoneofthemostimportantfunctionsinmathematics.Exponentialfunctionshave
manyscientificapplications,suchaspopulationgrowthandradioactivedecay.Exponentialfunctionarealso
usedinfinance,soifyouhaveacreditcard,bankaccount,carloan,orhomeloanitisimportantto
understandexponentialfunctionsandhowtheywork.
Exponentialfunctionsarefunctionwherethevariablexisin theexponent.Someexamplesofexponential
functionsaref(x)=2
x
,f(x)=5
x – 2
,orf(x)=9
2x+1
.Ineachofthethreeexamplesthevariablexisinthe
exponent,whichmakeseachof theexamplesexponentialfunctions.
GraphingExponentialFunctions
Tobegingraphingexponentialfunctionswewillstartwithtwoexamples.Wewillgraphthetwo
exponentialfunctionsbymakingatableofvaluesandplottingthepoints.Aftergraphingthefirsttwo
exampleswewilltakealookatthesimilaritiesanddifferencesbetweenthetwographs.
Whencreatingatableofvalues,Ialwayssuggeststartingwiththenumbersx=–2,–1,0,1,and2becauseit
isimportanttohavedifferenttypesofnumbers,somenegative,somepositive,andzero.
Example1:Graphf(x)=2
x
.
x x
f (x ) 2 =
–2 2
2
1 1
f ( 2) 2 4
2
-
-= = =
–1 1
1
1 1
f ( 1) 2 2
2
-
-= = =
0 0
f (0) 2 1 = =
1 1
f (1) 2 2 = =
2 2
f (2) 2 4 = =
Byplottingthefivepointsinthetableaboveandconnectingthepoints,wegetthegraphshownabove.
Noticethatasthexvaluesgetsmaller,x=–1,–2,etc.thegraphofthefunctiongetscloserandclosertothe
xaxis,butnevertouchesthexaxis.Thismeansthatthereisahorizontalasymptoteatthexaxisory=0.A
horizontalasymptoteisahorizontallinethatthegraphgetscloserandcloserto.
An ExponentialFunctionisafunctionoftheform
f(x)=b
x ory=b
x
wherebiscalledthe“base”andbisapositiverealnumberotherthan1(b>0andb ≠1).Thedomain
ofanexponentialfunctionisallrealnumbers,thatis,xcanbeanyrealnumber.
pf3

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Download Exploring Exponential Functions: Graphing and Identifying Growth or Decay and more Lecture notes Mathematics in PDF only on Docsity!

Graphing Exponential Functions

What is an Exponential Function?

Exponential functions are one of the most important functions in mathematics. Exponential functions have

many scientific applications, such as population growth and radioactive decay. Exponential function are also

used in finance, so if you have a credit card, bank account, car loan, or home loan it is important to

understand exponential functions and how they work.

Exponential functions are function where the variable x is in the exponent. Some examples of exponential

functions are f(x) = 2

x

, f(x) = 5

x – 2

, or f(x) = 9

2 x + 1

. In each of the three examples the variable x is in the

exponent, which makes each of the examples exponential functions.

Graphing Exponential Functions

To begin graphing exponential functions we will start with two examples. We will graph the two

exponential functions by making a table of values and plotting the points. After graphing the first two

examples we will take a look at the similarities and differences between the two graphs.

When creating a table of values, I always suggest starting with the numbers x = –2, –1, 0 , 1 , and 2 because it

is important to have different types of numbers, some negative, some positive, and zero.

Example 1 : Graph f(x) = 2

x

.

x

x

f (x) = 2

2

2

f ( 2 ) 2

  • = = =

1

1

f ( 1 ) 2

  • = = =

0

f ( 0 ) = 2 = 1

1

f ( 1 ) = 2 = 2

2

f ( 2 ) = 2 = 4

By plotting the five points in the table above and connecting the points, we get the graph shown above.

Notice that as the x values get smaller, x = –1, –2, etc. the graph of the function gets closer and closer to the

x axis, but never touches the x axis. This means that there is a horizontal asymptote at the x axis or y = 0. A

horizontal asymptote is a horizontal line that the graph gets closer and closer to.

An Exponential Function is a function of the form

f(x) = b

x

or y = b

x

where b is called the “base” and b is a positive real number other than 1(b > 0 and b ≠ 1). The domain

of an exponential function is all real numbers, that is, x can be any real number.

Example 2 : Graph

x

1

f (x)

Ê ˆ

Á ˜

Ë ¯

x

x

1

f (x)

Ê ˆ

Á ˜

Ë ¯

2 2 2

2 2

f ( 2 ) 4

Ê ˆ

Á ˜

Ë ¯

(^1 1 )

1 1

f ( 1 ) 2

  • (^) -

Ê ˆ

Á ˜

Ë ¯

0

1

f ( 0 ) 1

Ê ˆ

Á ˜

Ë ¯

1 1

1

f ( 1 )

Ê ˆ

Á ˜

Ë ¯

2 2

2

f ( 2 )

Ê ˆ

Á ˜

Ë ¯

By plotting the five points in the table above and connecting the points, we get the graph shown above.

Notice that as the x values get larger, x = 1, 2, etc. the graph of the function gets closer and closer to the x

axis, but never touches the x axis. This means that there is a horizontal asymptote at the x axis or y = 0.

Now we can look at the similarities and differences between the graphs.

Similarities

  • The domain for each example is all real numbers.
  • The range for each example is all positive real numbers.
  • Both graphs pass through the point ( 0 , 1 ) or the y intercept in each graph is 1.
  • Both graphs get closer and closer to the x axis, but do not touch the x axis. So, each graph has a

horizontal asymptote at the x axis or y = 0.

Differences

  • In Example 1, the graph goes upwards as it goes from left to right making it an increasing function.

An exponential function that goes up from left to right is called “Exponential Growth”.

  • In Example 2, the graph goes downwards as it goes from left to right making it a decreasing

function. An exponential function that goes down from left to right is called “Exponential Decay”.

Exponential Growth or Exponential Decay

If we are given an exponential function and asked to predict if the resulting graph would be exponential

growth or exponential decay, how can we correctly answer the question without actually drawing the graph?

The key to correctly answering the question is to look at the base of the exponential function. Consider the

following exponential functions and try to predict growth or decay by looking at the base of the function:

x

f (x)

Ê ˆ

Á ˜

Ë ¯

and

x

f (x)

Ê ˆ

Á ˜

Ë ¯